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{\bf Math 166: Honors Calculus II}
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{\bf Exam II}\quad {\sl Apr. 8, 1999}

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There are 6 questions, each with several parts, worth a total of 110 points.
Be sure to show all your work and justify all steps.
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\begin{enumerate}
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\item (20 pts) Define the following
  \begin{enumerate}[a)]
  \item $\lim_{x\to a} f(x) = \infty$

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  \item $\lim_{n\to \infty} a_n = L$

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  \item $f(x) = o(g(x))$ as $x \to a$.

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  \item $\sum_{n=1}^\infty a_n$ converges.

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  \end{enumerate}


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\item (20 pts) State the following theorems precisely.
  \begin{enumerate}[a)]

  \item  Taylor's Theorem with Integral Remainder.
  \answerspace

  \item State L'H\^opital's Theorem.

  \answerspace

  \item The Limit Comparison Test.

  \answerspace

  \item The Integral Test.

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  \end{enumerate}

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\item (20 pts)

  \begin{enumerate}[a)]
  \item Express $x^3\log(1+x)$ in $o$-notation; include at least three
  non-zero terms.

  \vfill
  \answerspace

  \item Simplify
  $\frac{1+x+x^2+o(x^3)}{1-x^2+x^3+o(x^3)}$ into the form
  $a_0 + a_1 x + a_2x^2 + o(x^2)$.

  \vfill
  \answerspace

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   \item Let $f(x) = \sin(x)$ and consider $f(x) = T_nf(x;0) + E_nf(x)$.
   Find $n$ such that $|E_nf(x)| < 10^{-3}$ for all~$x\in[-1,1]$.

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  \item Prove that if $|r|<1$ then $\sum_{n=0}^\infty r^n = \frac{1}{1-r}$.

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  \end{enumerate}

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\item (20 pts) Compute the following limits. Be sure to justify each step.
  \begin{enumerate}[a)]

  \item $\lim_{x\to 0}(1+h(x))^{1/h(x)}$ where $h(x)$ is a
  non-constant differentiable function in a neighborhood of~0 and $h(0) = 0$.

  \vfill
  \answerspace

  \item $\lim_{x\to 0} \frac{e^{x^2}-\cos(x)}{x^2}$

  \vfill
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  \item $\lim_{x\to\infty} \sqrt{2x+x^2} - \sqrt{x+x^2}$

  \vfill
  \answerspace

  \item $\lim_{n\to\infty}\frac{(n + (-1)^n)(n+1)}{3n^2}$ \quad ($n\in\N$)

  \vfill
  \answerspace

  \end{enumerate}


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\item (15 pts) Compute the sums of the following series.
  \begin{enumerate}[a)]

  \item $\sum_{n=0}^\infty \frac{a^{n} + (-1)^n}{b^{2n}}$ where $0 < a < b$

  \vfill
  \answerspace

  \item $\sum_{n=1}^\infty \frac{2n + 1}{n^2(n+1)^2}$

  \vfill
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  \item $\sum_{n=0}^\infty \frac{n}{n+1}x^n$ for $|x|<1$\newline
  (You may use integration or differentiation term by term.)  

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  \answerspace

  \end{enumerate}

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\item (15 pts) Test the following series for convergence. Justify your answers.
  \begin{enumerate}[a)]

  \item $\sum_{n=0}^\infty\frac{1}{n!}$

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  \answerspace

  \item $\sum_{n=1}^\infty\frac{\sqrt{n+1}}{n^2}$

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  \item $\sum_{n=2}^\infty\frac{1}{n\log(n)}$

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  \end{enumerate}


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